Course Title : Physics of metallic condensed matter

Lecture 3: Imperfections in metals

Lecture Plan:1. Point defects : vacancies and self-interstitials, impurities2 Solid solutions: specification of composition 2. Solid solutions: specification of composition 3. Dislocations (edge and screw dislocation)4. Interfacial defects : external surfaces, grain boundaries, twins5 Volume defects 5. Volume defects 6. Microscopic techniques for defect investigation (Basic concepts of

microscopy examination; optical microscopy)

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Introduction : Just before metallic solids were represented as perfectcrystals having ideal crystalline lattice without any defects. But such an idealizedsolid does not exist; all contain large numbers of various defects orimperfections.

Crystalline defect refers to a lattice irregularity having one or more of itsdimensions on the order of an atomic diameter Classification of crystallinedimensions on the order of an atomic diameter. Classification of crystallinedefects is frequently made according to geometry or dimensionality of the defect.There are point defects (i.e. associated with one or two atomic positions), linear(or one-dimensional) defects, and interfacial defects, or boundaries, which are

Vocabulory and special terms

two-dimensional and, at last, volume defects (3-dimensional ones).

Alloy -Atomic percent - atomic vibration Burgers vector

point defect scanning electron microscope(SEM)screw dislocation Burgers vector

Composition - dislocation line edge dislocation

screw dislocation self-interstitial substitutional solid solution g

grain size Imperfection - interstitial solid solution

solute - Solvent transmission electron microscope (TEM)Vacancy

Microstructure - Photomicrograph

Vacancy - weight percent

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I. Point defects : vacancies, self-interstitials, impurities

Two-dimensional representation of a vacancy and an interstitial

I. Point defects : vacancies, self interstitials, impurities

of a vacancy and an interstitial atom

A)Vacancy, or vacant lattice site, one normally occupied from which one normally occupied from which an atom is missing. All crystalline solids contain vacancies and, in fact, it is not possible to create such a it is not possible to create such a material that is free of these defects.

B) A self-interstitial defect is an )atom embedded into an interstitial site. An interstitial site is a void volume incide the unit cell.

Defect characteristics:Edef - the formation energy of a defect is the energy differencebetween a crystal with a defect and a perfect crystal containing thebetween a crystal with a defect, and a perfect crystal, containing thesame number of atoms. Vdef - the formation volume is the difference involume between these two crystals. 3

B. Types of self-interstitial voids

in BCC lattice : 3 octahedral and 6 tetrahedral interstitial voids/atom, in FCC and HCP lattices - 1 octa- and 2 tetrahedral interstitial voids/atom. Volumes of octahedral and tetrahedral voids for FCC and HCP: 0.41R, 0.22R; for BCC : 0,15R and 0.29R, correspondingly.

Positions of tetra- and octa- interstitial voids

a,b - FCC lattice ; c, d - HCP lattice; e f BCC latticee, f BCC lattice

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Around a point defect the lattice is distorted. Simple way a point defect can

Distortions around point defectsAround a point defect the lattice is distorted. Simple way a point defect can

be represented in elastic environment as a center of compression (for a vacancy) or expansion (for an interstitial). Stresses and strains around such center diminish inversely to the third power of distance from it. It means that essential t i di l t l t ll di t f t t i di t atomic displacements are only at small distances of one-two atomic diameters

from point defects. This area is named as a defect core.

Consideration pair atomic interactions allows to Consideration pair atomic interactions allows to evaluate that in FCC lattice for a vacancy the atoms of 1-st coordination sphere are displaced on - 0.84% of interatomic distance in the direction to d f h 2 d 0 25% f d f h 3 d

Around an interstitial atom neighbors in the 1-st di ti h di l d f it 20%

defect, the 2-nd, +0.25% from defect, the 3-d -0.03%.

coordination sphere are displaced from it on ~20% of interatomic distance but in the 2-nd layer displaced to it.

In metals, a self-interstitial introduces relatively large distortions in thesurrounding lattice because the atom is substantially larger than theinterstitial position in which it is situated. Consequently, the formation of thisdefect is not highly probable and it exists in very small concentrations whichdefect is not highly probable, and it exists in very small concentrations, whichare significantly lower than for vacancies. Theoretically, the formation energyof vacancy (~1eV) is much less than self-interstitial atom (3-4eV). 5

on - sites occupied with atoms + overall number of on - empty sites

nnno =+ overall number of sites in the latticeVacancy contribution to free energy of a real crystal is termed as

TSnEF f = ( ) ( )( ))()( nnnnnnnnnnnkS oooo lll ++=wherefE

k is the Boltzmanns constant. k =1.3810-23 J/(atom K) =8.62 10-5 eV/(atom K)

fE - the formation energy of a vacancy defect per atom. In case of

equilibrium state F reaches minimum

0dF 0 nkTE l0=

dn0=

++ nn

nnkTEo

f l

nno >>

=

kTE

nn fov

expo

i.e. the equilibrium number of vacancies for a given quantity of material depends on and increases with temperature according to exponential

kTovp

depends on and increases with temperature according to exponential dependency. For most metals, the fraction of vacancies nV/no just below the melting temperature is on the order of 10-4; that is, one lattice site out of 10,000 will be empty.

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The energy barrier that an atom must The energy barrier that an atom must overcome to jump into a vacancy.

mEIn sites of a crystalline lattice atoms oscillate with vibration energy

om qE > there is a nonzero probability for atom to jump from its position into the nearest vacant site.

Formation of vacancy defect in solid in a surface layer7

= EAv m

exp A number of atoms jumps into a vacancy per a d i

activation energy for vacancy migration per atom)

mE

=kT

Avm exp second. A- is constant

g p )

For example in copper A 1015, mE ~29 kcal/mole then

vm = 3 1010 Jumps/c at 1350 K and vm = 106 Jumps/c at 300 K.

n

+ EEA mf

In average over a crystal containing vacancies the rate of atomic or

vacancy jumps is

=kT

Av mfa exp

.Table 1 Experimental values of formation and migration energies for vacancyTable 1 Experimental values of formation and migration energies for vacancy

Cu Ag Au Al Ni Pt

f , 1,14 1,1 0,98 0,76 1,4 1,51m , 1,08 0,83 0,82 0,65 1,5 1,38

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IMPURITIES IN SOLIDSIMPURITIES IN SOLIDS

A pure metal consisting of only one type of atom just isnt possible;impurity or foreign atoms will always be present and some will exist asimpurity or foreign atoms will always be present, and some will exist ascrystalline point defects. Even with sophisticated techniques, it is difficult torefine metals to a purity in excess of 99.9999%. At this level, on the order of1022 to 1023 impurity atoms will be present in 1m3 of material. When crystal isp y p ycomposed from some constituents (metallic on nonmetallic element) it istermed as alloy. Addition of impurity atoms to a metal will result in the formation of asolid solution and/or a new second phase depending on the kinds ofsolid solution and/or a new second phase, depending on the kinds ofimpurity, their concentrations, and the temperature of the alloy. With regard to alloys, solute and solvent are terms that are commonlyemployed. Solvent represents the element or compound that is present in thep y p p pgreatest amount; on occasion, solvent atoms are also called host atoms.Solute is used to denote an element or compound present in a minorconcentration. Impurity point defects are found in solid solutions of which there are two Impurity point defects are found in solid solutions, of which there are twotypes: substitutional () and interstitial (). For thesubstitutional type, solute or impurity atoms replace or substitute for the hostatoms.

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IMPURITIES IN SOLIDS

Host atoms Substitutional impurity atoms

(Solute presented in minor quantity )(Solvent presented in great quantity )

Host atoms Substitutional impurity atoms

Interstitial impurity atoms

Pure substanceSubstitutional solid solution

Interstitial solid solution

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The composition (or concentration) of an alloy is expressed in terms of itsconstituent elements. The two most common ways to specify composition arey p y pweight (or mass) percent and atom percent. The basis for weight percent(wt%) is the weight of a particular element relative to the total alloy weight. Foran alloy that contains two hypothetical atoms denoted by 1 and 2, theconcent ation of 1 in t% C1 is defined as

10011 += mmmC

concentration of 1 in wt%, C1, is defined as

where m1 and m2 represent the weight (or mass)of elements 1 and 2 respectively

21 + mm of elements 1 and 2, respectively.The basis for atom percent (at%) calculations is the number of moles of anelement in relation to the total moles of the elements in the alloy. The numberyof moles in some specified mass of a hypothetical element 1, nm1, may becomputed as follows:

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'mn =1

1 Anm

m1 and A1 denote the mass (in grams) and atomic weight, respectively, for element 1 Concentration in terms of atom percent of element 1 in an alloy element 1.Concentration in terms of atom percent of element 1 in an alloy containing element 1 and element 2 atoms, is defined by

100' 1 mnC 100'21

11 += mm

m

nnC

11

100' 21ACC 100' 12ACC100'' 21 =+CC

Composition Conversions

from wt.% into at.% 100'

1221

211 += ACACC 100' 1221

122 += ACACC

'AC 'AC

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from at% into wt.% 100''

'

2211

111 += ACAC

ACC 100''

'

2211

222 += ACAC

ACC10021 =+CC

from wt.% to mass per unit volume

Computation of density (f 2 l t

Computation of atomic i ht (f t l t per unit volume (for a 2-element

metal alloy)weight (for a two-element

metal alloy)12

DISLOCATIONSLINEAR DEFECTS

A dislocation is a linear or one dimensional defect around which some of theA dislocation is a linear or one-dimensional defect around which some of theatoms are misaligned. For example, an extra portion of a plane of atoms, or half-plane, the edge of which terminates within the crystal.

This is termed an edge dislocation; it is a This is termed an edge dislocation; it is a linear defect that centers on the line that is defined along the end of the extra half-plane of atoms. This is sometimes termed the dislocation line, which, for the edge dislocation is perpendicular to the plane of the page .Within the region around the dislocation line there is some localized lattice distortion line there is some localized lattice distortion. The atoms above the dislocation line are squeezed together, and those below are pulled apart; this is reflected in the slight curvature

The atom positions around an edge

di l ti t h lf

for the vertical planes of atoms as they bend around this extra half-plane. The magnitude of this distortion decreases with distance away from the dislocation line; at positions far dislocation; extra half-

plane of atomsshown in perspective.

from the dislocation line; at positions far removed, the crystal lattice is virtually perfect. Sometimes the edge dislocation is represented by the symbol which also indicates the position of the dislocation line .

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Another type of dislocation, called a screw dislocation, may be thought of as being formed by a shear stress that is applied to produce the distortion : the

f t i f th t l i hift d t i di t t th i ht upper front region of the crystal is shifted one atomic distance to the right relative to the bottom portion. The atomic distortion associated with a screw dislocation is also linear and along a dislocation line, line AB. The screw dislocation derives its name from the spiral or helical path or ramp that is dislocation derives its name from the spiral or helical path or ramp that is traced around the dislocation line by the atomic planes of atoms. Sometimes the symbol is used to designate a screw dislocation

(a) A screw dislocation within a crystal. (b) The screw dislocation in (a) as viewed from above. The dislocation line extends along line AB. Atom positions above the slip plane are designated by open circles those below by solid circles Most dislocaslip plane are designated by open circles, those below by solid circles. Most disloca-tions found in crystalline materials are probably neither pure edge nor pure screw, but exhibit components of both types; these are termed mixed dislocations.. 14

A shift of the upper part of the crystal relatively the bottom

A=F b = b l1l2

The analogy for dislocation moving

1 2l1 length of crystal; l2 width

gy gThe velocity of dislocation moving can change from low to very high values 10-7- 105cm/s

The magnitude and direction of the lattice distortion associated with a lattice distortion associated with a dislocation is expressed in terms of a Burgers vector, denoted by a b.

Burgers vector of an edge and screw dislocations 15

Burgers vector features :

1. The nature of a dislocation (i.e., edge, screw, or mixed) is defined by the relative orientations of dislocation line and Burgers vector. For an edge, they are perpendicular , whereas for a screw, they are parallel; they are neither perpendicular nor parallel for a mixed dislocation.

2. Even though a dislocation changes direction and nature within a crystal (e.g., from edge to mixed to screw), the Burgers vector will be the same at all points along its line For example all positions of the curved dislocation have the points along its line. For example, all positions of the curved dislocation have the Burgers vector shown.

3. Burgers vectors of nondislocation defects are equal zero.4. Burgers vector and a dislocations line uniquely determine a slip plane. For metallic materials, the Burgers vector for a dislocation will point in a

close-packed crystallographic direction and will be of magnitude equal to the interatomic spacing.

222zyx bbbb ++=The length of Burgers vector is equaled

Burgers vector length

][ wunab =r 222 wunab ++= For a cubic crystal

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Burgers vector of a perfect dislocation

Dislocation reactions

allows identical translation of lattice. Forexample, Burgers vectors in primitive cubiclattice

100a 110aDislocations whose Burgers vector is not avector of identical translation are termedas partial dislocations:

100a 110a

1113/a 1216/aas partial dislocations:

Perfect and partial dislocations can interact with each other.The rule for dislocation reaction : sum of Burgers vectors of initial

dislocations must be equal to the sum of Burgers vectors after reaction.Energy Frank criterium: the reaction is possible if the sum of squaredEnergy Frank criterium: the reaction is possible if the sum of squared

Burgers vectors of initial dislocations is greater than the sum of squares vectorsof resulted dislocations. It means that a reaction must lead to decrease ofenergy of dislocation system.

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22

21 bbb +>

bbb

gy y

the dislocation can dissociates into two dislocations andthe reaction

321 bbb += is possible.17

lGbED 2 = Dislocation elastic energy , where = 0.5 , 1.0.

G shear modulus b Burgers vector l dislocation length

= bF A force on a unit of dislocation length is equaled to the product of Burgers vector to a tangential stress in the slip plane.

G shear modulus, b - Burgers vector, l dislocation length

lbF l is a circle ark on which the F affects in the OA direction This force moves the dislocation line in the slip plane acrossthe crystal and it is perpendicular to the dislocation line.

o u g o o a a g a p p a

lbFext =)2/sin(2 TF =

l - is a circle ark on which the Fext affects in the OA direction.

where T is a line tension of an ark defined as

- the counteracting forceF

2GbT =2/)2/sin(

where T is a line tension of an ark, defined as

In case of small angles

Fext

)(

,TF = rl / = rlTF /=lGb /2

g

rlGbF /2=lbFF ext == rGb / =In case

Under tangential stresses the dislocation line curves

- the stress necessary to curve the dislocation line to arc-like shape.

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Dislocation density

V

l= [cm-2 ] . Dislocation density is a summary lengths of alldislocation lines in a unit volume. Dislocations can be observedin crystalline materials using electron-microscopic techniques..

High-magnification transmission electron micrograph, the dark lines are the dislocations

To measure the linear-intercept method is used.

+=

2

2

1

1

LN

LN

tM LL

dIn micrographs the number of intersection of dislocations NL1 , NL2 with arbitrary horizontal and vertical lines with the length L1 and L2, respectively; M is magnification; t is the thickness of a foil investigated

21

is magnification; t is the thickness of a foil investigated.

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INTERFACIAL DEFECTS

Interfacial defects are boundaries that have two dimensions and normally separate regions of the materials that have different crystal structures and/or crystallographic orientations These imperfections include external surfaces crystallographic orientations. These imperfections include external surfaces, grain boundaries, phase and twin boundaries.

Grain Boundaries

Grain boundary is a boundary separating t ll i t l h i diff t two small grains or crystals having different crystallographic orientations in polycrystalline materials. Within the boundary region, which is probably just y g , p y jseveral atom distances wide, there is some atomic mismatch in a transition from the crystalline orientation of one grain to that of

dj t Schematic diagram showing small and high-angle grain

boundaries and the

an adjacent one. Various degrees of crystallographic

misalignment between adjacent grains are possible.

adjacent atom positions.p

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When this orientation mismatch is slight (i.e. less than 10-15o), on the order of a few degrees, then the term small- (or low-) angle grain boundary is used. These boundaries can be described in terms of dislocation arrays. This type is called a tilt boundary; the angle of misorientation, , is also indicated in the figure. The small angle symmetric tilt boundary is formed by a wall of edge dislocations of the same sign with parallel both Burgers vectors and slip planes dislocations of the same sign with parallel both Burgers vectors and slip planes. When the angle of misorientation is parallel to the boundary, a twist boundary ( ) results, which can be described by an array of screw dislocations.

However, the most common are high angle boundaries having great angle misorientation between neighbor grains. The equilibrium linear defects of such high angle boundary are grain boundary dislocations which serve accommodation role in adjustment of two crystals accommodation role in adjustment of two crystals.

Small angle grain g gboundary and dislocation wall

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Phase Boundaries

Phase boundaries exist in multiphase materials wherein a different phase Phase boundaries exist in multiphase materials , wherein a different phase exists on each side of the boundary; furthermore, each of the constituent phases has its own distinctive physical and/or chemical characteristics. Phase boundaries are classified as coherent with full conjugation between two lattices; semicoherent one with misfit dislocations and incoherent one.

Semicohe ent bo ndaCoherent boundary Semicoherent boundary

Incoherent boundary

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Twin Boundaries

A t i b d i i l t f i b d hi h th i ifi A twin boundary is a special type of grain boundary across which there is a specific mirror lattice symmetry; that is, atoms on one side of the boundary are located in mirror-image positions of the atoms on the other side. The region of material between these boundaries is appropriately termed a twin. Twins result from atomic between these boundaries is appropriately termed a twin. Twins result from atomic displacements that are produced from applied mechanical shear forces (mechanical twins), and also during annealing heat treatments following deformation (annealing twins). Twinning occurs on a definite crystallographic plane

d i ifi di ti b th f hi h d d th t l t t and in a specific direction, both of which depend on the crystal structure. Annealing twins are typically found in metals that have the FCC crystal structure, whereas mechanical twins are observed in BCC and HCP metals.

Annealing twins and deformation twins

30mkm

Schematic diagram of a twin plane

Annealing twins and deformation twinsin steel

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These include void or gas pores cracks foreign inclusions They are

Volume defects :

These include void or gas pores, cracks, foreign inclusions. They are normally introduced during processing and fabrication steps.

voids precipitates cracks

Defects of T-type are characterized by the average density and the i daverage size d

where M magnification; S area of a photograph; t thickness of a specimen;

N

jdiNM

k

i21 photograph; t-thickness of a specimen; Ni number of defects in S area; k number of photographsi

jj

i MNd

== 1Stk

i== 1124

Microscopic techniques for defect investigation (Basic concepts of microscopy examination)

Optical, electron scanning and transmission microscopes aid ininvestigations of the microstructural features of all material types. Thesecan use photographic equipment for image recording (the photograph onwhich the image is recorded is called a photomicrograph) or computergenerated images. Microscopic examination is an extremely useful tool in the studyand characterization of materialsand characterization of materials. Important applications of microstructural examinations :

- to ensure that the associations between the properties andstructure (and defects) are properly understood,

- to predict the properties of materials once these relationships havebeen established,

- to design alloys with new property combinations,- to determine whether a material has been correctly heat-treatedto determine whether a material has been correctly heat treated,- to ascertain the mode of mechanical fracture.

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(a) Polished and etched grains as they might appear when

Optical microscopy

as they might appear when viewed with an optical microscope. (b) Section taken through these grains showing how the etching how the etching characteristics and resulting surface texture vary from grain to grain because of differences in crystallographic differences in crystallographic orientation. (c) Photomicrograph of a polycrystalline brass specimen specimen.

(a) Section of a grain boundary and its surface boundary and its surface groove produced by etching; the light reflection characteristics in the vicinity of the groove are also shown. of the groove are also shown. (b) Photomicrograph of the surface of a polished and etched polycrystalline specimen of an ironspec e o a ochromium alloy in which the grain boundaries appear dark. 26

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